36. Sufficient statistics with nuisance parameters. (i) A statistic T is said to be partially sufficient for 8 in the presence of a nuisance parameter 1/ if the parameter space is the direct product of the set of possible 8- and 1/-values, and if the following two conditions hold:

(a) the conditional distribution given T = t depends only on 1/;

(b) the marginal distribution of T depends only on 8. If these conditions are satisfied, there exists a UMP test for testing the composite hypothesis H : 8 = 80 against the composite class of alternatives 8 = 81, which depends only on T. (ii) Part (i) provides an alternative proof that the test of Example 8 is UMP. [Let I/Io(t) be the most powerful level a test for testing 80 against 81 that depends only on t, let 1/1 (x) be any level-a test, and let 1/1 (t) = E~l [ 1/1 (X) It]. Since E9,I/I(T) = E9,. ~ll/l( X), it follows that 1/1 is a level-a test of H and its power, and therefore the power of 1/1, does not exceed the power of %.] Note . For further discussion of this and related concepts of partial sufficiency see Dawid (1975), Sprott (1975), Basu (1978), and Barndorff-Nielsen (1978).